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Its meant to just be a joke that his real name is also a term of endearment, setting up some good lines from the others, not that it is a nickname. To a British ear, Darling sounds convincingly like it could be a last name of someone from the upper classes - like an officer in the British Army at the time.

Its worth pointing out that Darling is definitely more likely a last name than Blackadder itself. For example Darling was the last name of the family in Peter Pan, a story from the same era.